Sub-Resolution Assist Features

ABSTRACT

Methods of semiconductor device fabrication are provided. In an embodiment, a method of semiconductor device fabrication includes receiving a first mask design comprising a first mask function, determining a transmission cross coefficient (TCC) of an exposure tool, decomposing the TCC into a plurality orders of eigenvalues and a plurality orders of eigenfunctions, calculating a kernel based on the plurality orders of eigenvalues and the plurality orders of eigenfunctions; and determining a first sub-resolution assist feature (SRAF) seed map by convoluting the first mask function and the kernel.

PRIORITY DATA

This application claims the benefit of U.S. Provisional Application No.62/877,437, filed Jul. 23, 2019, the entirety of which is herebyincorporated by reference herein.

BACKGROUND

The electronics industry has experienced an ever-increasing demand forsmaller and faster electronic devices to support a greater number ofincreasingly complex and sophisticated functions. Accordingly, there isa continuing trend in the semiconductor industry to manufacturelow-cost, high-performance, and low-power integrated circuits (ICs).Thus far these goals have been achieved in large part by scaling downsemiconductor IC dimensions (e.g., minimum feature size) and therebyimproving production efficiency and lowering associated costs. However,such scaling down has also introduced increased complexity to thesemiconductor manufacturing process. Thus, the realization of continuedadvances in semiconductor ICs and devices calls for similar advances insemiconductor manufacturing processes and technology.

As merely one example, scaling down of IC dimensions has been achievedby extending the usable resolution of a given lithography generation bythe use of one or more resolution enhancement technologies (RETs), suchas phase shift masks (PSMs), off-axis illumination (OAI), opticalproximity correction (OPC), and insertion of sub-resolution assistfeatures (SRAFs) into a design layout. Several SRAF insertion orplacement techniques have been proposed. Some of them, being rule-based,have relatively short turn-around time but far-from-ideal accuracy. Someof them use numerous iterations of mask optimization to achieveoutstanding accuracy but take a long time for each SRAF insertionexercise. Thus, existing techniques have not proved entirelysatisfactory in all respects.

BRIEF DESCRIPTION OF THE DRAWINGS

Aspects of the present disclosure are best understood from the followingdetailed description when they are read with the accompanying figures.It is noted that, in accordance with the standard practice in theindustry, various features are not drawn to scale. In fact, thedimensions of the various features may be arbitrarily increased orreduced for clarity of discussion.

FIG. 1 is a flow chart of an embodiment of a method of semiconductordevice fabrication, according to various aspects of the presentdisclosure.

FIG. 2 is a schematic illustration of process steps of the method inFIG. 1, according to various aspects of the present disclosure.

FIG. 3 is a schematic illustration of two points on a mask, according tovarious aspects of the present disclosure.

FIG. 4 is a schematic illustration of components of an exposure tool,according to various aspects of the present disclosure.

FIG. 5 is a flow chart of an embodiment of a method of semiconductordevice fabrication, according to various aspects of the presentdisclosure.

FIG. 6 is a schematic illustration of process steps of the method inFIG. 5, according to various aspects of the present disclosure.

FIG. 7 is a schematic illustration of diffraction of illumination at anideal mask, according to various aspects of the present disclosure.

FIG. 8 is a schematic illustration of diffraction of illumination at areal-world mask with a thickness, according to various aspects of thepresent disclosure.

FIG. 9 is a flow chart of an embodiment of a method of semiconductordevice fabrication where mask three-dimensional (3D) effect isconsidered, according to various aspects of the present disclosure.

FIG. 10 is a schematic illustration of process steps of the method inFIG. 9, according to various aspects of the present disclosure.

DETAILED DESCRIPTION

The following disclosure provides many different embodiments, orexamples, for implementing different features of the provided subjectmatter. Specific examples of components and arrangements are describedbelow to simplify the present disclosure. These are, of course, merelyexamples and are not intended to be limiting. For example, the formationof a first feature over or on a second feature in the description thatfollows may include embodiments in which the first and second featuresare formed in direct contact, and may also include embodiments in whichadditional features may be formed between the first and second features,such that the first and second features may not be in direct contact. Inaddition, the present disclosure may repeat reference numerals and/orletters in the various examples. This repetition is for the purpose ofsimplicity and clarity and does not in itself dictate a relationshipbetween the various embodiments and/or configurations discussed.

Further, spatially relative terms, such as “beneath,” “below,” “lower,”“above,” “upper” and the like, may be used herein for ease ofdescription to describe one element or feature's relationship to anotherelement(s) or feature(s) as illustrated in the figures. The spatiallyrelative terms are intended to encompass different orientations of thedevice in use or operation in addition to the orientation depicted inthe figures. The apparatus may be otherwise oriented (rotated 90 degreesor at other orientations) and the spatially relative descriptors usedherein may likewise be interpreted accordingly.

The present disclosure is generally related to methods of semiconductordevice fabrication. More specifically, the present disclosure is relatedto methods for generating an SRAF seed map for SRAF placement. SRAFs aremask features that are small enough not to be printed on a substrate(wafer) in a photolithography process but are so shaped and placed on amask to improve the quality of photolithography image on the substrate.Methods to determine the shapes and positions of SRAFs are thereof ofheightened interest. With respect to placement of SRAFs, several SRAFplacement techniques have been proposed. For example, a conventionalSRAF placement technique is a rule-based SRAF placement method. In thismethod, numerous test patterns and corresponding wafer images areobtained to populate empirical data and the empirical data is studiedand analyzed to establish the rules. SRAFs are then placed on a maskbased on such rules. Because SRAFs are placed based on a rule table, itsturn-around-time is short. However, because the test patterns may not berepresentative of the actual patterns, rule-based SRAF placementtechniques suffer from unsatisfactory accuracy.

Another conventional SRAF placement technique is inference mappinglithography (IML). Real-world exposure tools use partially coherentradiation source and their partial coherence may be decomposed in sum ofcoherent systems (SOCS) by performing decomposition on the transmissioncross coefficient (TCC). In terms of optical physics, the TCC representsautocorrelation of the radiation source of the exposure tool with theprojection pupil of the exposure tool. Therefore, the TCC is amathematical representation of the imaging capability of the exposuretool which includes an ensemble of various exposure conditions of theexposure tool. The TCC may be decomposed into a set of eigenfunctions(φ) and a set of eigenvalues (λ). IML only considers the first ordereigenfunction of the TCC to determine SRAF placement. Because only thefirst order eigenfunction is included in IML, effect of exposureconditions of the exposure tool may not be sufficiently factored andaccuracy may be less than satisfactory.

Still another conventional SRAF placement technique is called inverselithography technology (ILT). ILT received its name due to its approachto lithography in an inverse fashion. Instead of calculating the aerialimage based on a given mask design, it calculates a mask designnecessary to generate a target aerial image. Although ILT may havesuperior accuracy, its turn-around-time may be unduly long andintractable. In some instances, ILT may require more than 300 times ofthe time needed to conclude a rule-based SRAF placement process. That iswhy ILT is currently mostly used to perform spot repairs of mask.Moreover, while mask three-dimensional (3D) effect has been observed andresearched, integration of the mask 3D effect has been largely absentfrom conventional SRAF placement techniques.

The present disclosure put forth methods of semiconductor devicefabrication where the placement of SRAFs includes better considerationof exposure conditions of the exposure tool and influence due to mask 3Deffect. By including all order of the eigenvalues and eigenfunctions ofTCC in the calculation of a kernel, methods according to embodiments ofthe present disclosure consider exposure conditions of the exposuretool, including illumination intensity of the exposure tool, a numericalaperture of the exposure tool, a depth of focus (DOF), a thickness of aresist stack to be patterned, or a range of an aberration. In addition,methods of the present disclosure may include diffraction components toaddress polarization due to mask three-dimensional (3D) effect. Becauseof the consideration of the exposure conditions and the mask 3D effect,methods of the present disclosure have better accuracy than conventionalrule-based SRAF placement techniques and IML techniques. Moreover, whenmethods of the present disclosure are used, a kernel generated based ona set of exposure conditions of an exposure tool may be stored andreused whenever a new mask design is introduced. The reuse of the kernelmay greatly reduce the turn-around-time to a level similar to that ofthe rule-based techniques, which is a fraction of the turn-around-timeof ILT. In some instances, after a kernel of the exposure tool isgenerated and stored, the turn-around-time to generate an SRAF seed mapusing the methods of the present disclosure is about 10% or 50% more ofthat of the conventional rule-based techniques for SRAF seed mapgeneration.

IC manufacturing includes multiple entities, such as a design house, amask house, and an IC manufacturer (i.e., a fab). These entitiesinteract with one another in the design, development, and manufacturingcycles and/or services related to manufacturing an integrated circuit(IC) device. These entities are connected by a communications network,which may be a single network or a variety of different networks, suchas an intranet and the Internet, and may include wired and/or wirelesscommunication channels. Each entity may interact with other entities andmay provide services to and/or receive services from the other entities.One or more of the design house, mask house, and IC manufacturer mayhave a common owner, and may even coexist in a common facility and usecommon resources. In various embodiments, the design house, which mayinclude one or more design teams, generates an IC design layout. The ICdesign layout may include various geometrical patterns designed for thefabrication of the IC device. By way of example, the geometricalpatterns may correspond to patterns of metal, oxide, or semiconductorlayers that make up the various components of the IC device to befabricated. The various layers collectively form various features of theIC device. For example, various portions of the IC design layout mayinclude features such as an active region, a gate electrode, source anddrain regions, metal lines or vias of a metal interconnect, openings forbond pads, as well as other features known in the art which are to beformed within a semiconductor substrate (e.g., such as a silicon wafer)and various material layers disposed on the semiconductor substrate. Invarious examples, the design house implements a design procedure to formthe IC design layout. The design procedure may include logic design,physical design, and/or place and route. The IC design layout may bepresented in one or more data files having information related to thegeometrical patterns which are to be used for fabrication of the ICdevice. In some examples, the IC design layout may be expressed in aGDSII file format or DFII file format.

In some instances, the design house may transmit the IC design layout tothe mask house, for example, via the network connection described above.The mask house may then use the IC design layout to generate a maskdesign, such as the first mask design 202, modify the mask design toform a modified mask design, and manufacture one or more masks to beused for fabrication of the various layers of the IC device according tothe modified mask design. In various examples, the mask house performsmask data preparation, where the IC design layout is translated into aform that can be physically written by a mask writer, and maskfabrication, where the design layout prepared by the mask datapreparation is modified to generate a modified mask design and is thenfabricated. In some embodiments of the present disclosure, some of theoperations described above are not performed by the mask house, but theIC manufacturer, especially when information of the exposure tool isused.

FIG. 1 is a flow chart of an embodiment of a method 100 of semiconductordevice fabrication. Method 100 will be described below in conjunctionwith FIG. 2, which is a schematic illustration of process steps ofmethod 100. Additional steps can be provided before, during, and aftermethod 100, and some of the steps described can be moved, replaced, oreliminated for additional embodiments of method 100.

Referring now to FIGS. 1 and 2, the method 100 includes a block 102where a first mask design 202 is received. In some instances, the method100 is performed by the IC manufacturer and the first mask design 202 isreceived from the mask house. In some other instances, the first maskdesign is provided by the mask house and the method 100 is alsoperformed by the mask house. The first mask design 202 includes maskfeatures that can be characterized by or expressed in a first maskfunction (a¹(x, y)). Reference is now made to FIG. 3, which illustratesthe type of information described in a mask function, such as the firstmask function. A mask function contains relative positions andinteractions between a point (x,y) and a point (x-x′, y-y′), which isshifted from the point (x,y) by amount −x′ and −y′. Accordingly, thefirst mask function (a¹(x, y)) may also be expressed as two δ functions:

a ¹(x, y)=δ(x, y)+δ(x−x′, y−y′)

In some embodiments, the radiation of the exposure tool 204 may bepolarized and the polarization may be changed by the mask. For example,the radiation of the exposure tool 204 incident on the mask may bepolarized in the X direction and the light diffracted by the mask may bepolarized in the Y direction at the pupil. With respect to such a nearfield incoming and outgoing radiation pair, the first mask functionincludes an X-Y component (a^(1xy)(x, y)) and the X-Y componentrepresents a simulated interaction between the X-polarized radiation onthe mask and the Y-polarization radiation on the pupil. Similarly, withrespect to the X-polarized incident radiation and X-polarized outgoingradiation, the first mask function includes an X-X component (a^(1xx)(x,y))); with respect to the Y-polarized incident radiation and X-polarizedoutgoing radiation, the first mask function includes a Y-X component(a^(1yx)(x, y)); and with respect to the Y-polarized incident radiationand Y-polarized outgoing radiation, the first mask function includes aY-Y component (a^(1yy)(x, y)). In cases where the first mask design 202is assumed to be implemented as an ideal mask (shown as 702 in FIG. 7,described below), the X-X, X-Y, Y-X, and YY components are identical toone another. In cases where the first mask design 202 is assumed to beimplemented as a real-world mask (shown as 802 in in FIG. 8, describedbelow) with mask three-dimensional (3D) effect, the X-X, X-Y, Y-X, andYY components are not identical and should be considered separately.

Referring now to FIG. 1, the method 100 includes a block 104 where anexposure tool 204 is provided. The exposure tool 204 includes aradiation source or a light source. The exposure tool 204 may be a deepultraviolet (DUV) exposure tool with a DUV radiation sources (such asKrF excimer laser or ArF excimer laser) or an extreme ultraviolet (EUV)exposure tool with an EUV radiation sources (such as a tin droplet laserplasma EUV generation). In some embodiments, the exposure tool 204 maybe a DUV exposure tool with immersion lithography capabilities. Inaddition to a radiation source, the exposure tool 204 may includevarious optical components such as lens and mirrors and may beconfigured to expose a thickness of a resist stack on a substrate on asubstrate stage. The radiation source, optical components, andconfigurations of the exposure tool 204 may be characterized by a setexposure conditions, including, for example, illumination intensity ofthe radiation source of the exposure tool, shape of the radiationsource, a numerical aperture (NA) of the exposure tool, a depth of focus(DOF), a thickness of a resist stack to be patterned, a range of anaberration, and an amount of defocus. Unless the configuration of theexposure tool 204 is intentionally or inadvertently changed, the set ofexposure conditions is unique to the exposure tool 204. As will bedescribed in more details below, the unique set of exposure conditionsmakes it possible to only calculate kernel once and store the kernel forreuse with different mask designs.

Referring to FIGS. 1 and 2, the method 100 includes a block 106 where atransmission cross coefficient (TCC) 206 of the exposure tool 204 isdetermined based on the set of the exposure conditions. FIG. 4illustrates major component planes in an exposure tool (or lithographysystem). The exposure tool includes a radiation source (light source), amask (reticle), a pupil plane, and an aerial image. The radiation source(or light source) may be expressed as a function S (S(f, g)), where fand g are coordinates on a plane at the radiation source. As describedabove, the reticle (or the mask) may be expressed as a function a (a(x,y)), where x and y are coordinates on a plane at the mask. Radiationdiffracted by features on the mask may be expressed a Fourier Transformof the mask function:

â(f, g)=FT[a(x, y)]

At the pupil plane, a pupil function is represented as a function of P(P(f, g)). A complex conjugate of the pupil function is given by afunction of P* (P*(f, g)). The TCC is obtained by:

TCC(f ₁ , f ₂ , g ₂)=∫S(f, g)P(f+f ₁ , g+g ₁)P*(f+f ₂ , g+g ₂)dfdg

The aerial image may be expressed as a function of I, expressed as:

${I\left( {x,y} \right)} = {\int{\overset{}{\underset{}{\int{\int\int}}}{{TCC}\left( {f_{1},g_{1},f_{2},g_{2}} \right)}{\hat{a}\left( {f_{1},g_{1}} \right)}\hat{a}*\left( {f_{2},g_{2}} \right)^{{- i}\; 2\; {\pi\lbrack{({{({f_{1} + f_{2}})}_{x} + {{({{g\; 1} - {g\; 2}})}y}}\rbrack}}}{df}_{1}d\; g_{1}{df}_{2}d\; g_{2}}}$

Referring now to FIG. 1, the method 100 includes a block 108 where atransmission cross coefficient (TCC) 206 is decomposed into a pluralityorders of eigenvalues and a plurality orders of eigenfunctions. In someimplementations, through SOCS, the TCC 206 may be decomposed into aplurality orders of eigenvalues (each or which may be represented asλ_(i), where i denotes i-th eigenvalue of the TCC 206) and a pluralityorders of eigenfunctions (each of which may be represented as φ_(i),where i denotes i-th eigenfunction of the TCC 206). In some embodiments,there are N eigenfunctions and N eigenvalues, where N is an integer andmay be between 1 and the number of point sources in the light source.The TCC 206 may be rewritten as:

T(f ₁ , g ₁ , f ₂ , g ₂)=Σ_(i=1) ^(N)λ_(i)Φ_(i)(f ₁ ,g ₁) Φi*(f ₁ , g₁), where φ_(i) =FT[Φ _(i)(f , g)]

Referring to FIGS. 1 and 2, the method 100 includes a block 110 where akernel 208 is calculated based on the plurality orders of eigenvalues(λ_(i)) and the plurality orders of eigenfunctions (φ_(i)). The kernel208 includes interaction terms, each accounting for interaction betweenan origin point (0,0) and a point (x′,y′) on a plane of the mask. Thekernel 208, which includes only the real part (Re), may be written as:

${\Omega \left( {x^{\prime},y^{\prime}} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{{\lambda_{i}\left\lbrack {{\varphi_{i}\left( {x,y} \right)} \otimes {\delta \left( {x,\ y} \right)}} \right\rbrack}^{*}\left\lbrack {{\varphi_{i}\left( {x,y} \right)} \otimes {\delta \left( {{x - x^{\prime}},{y - y^{\prime}}} \right)}} \right\rbrack}} \right\rbrack}$     or$\mspace{79mu} {{\Omega \left( {x^{\prime},y^{\prime}} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{*}\left( {0,0} \right)}{\varphi_{i}\left( {{- x^{\prime}},{- y^{\prime}}} \right)}}} \right\rbrack}}$

If the coordinate representation (x′,y′) is changed to the coordinaterepresentation (x, y), the kernel 208 may be expressed as:

${\Omega \left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{*}\left( {0,0} \right)}{\varphi_{i}\left( {{- x},\ {- y}} \right)}}} \right\rbrack}$

Referring to FIG. 1, the method 100 includes a block 112 where thekernel 208 is stored in a memory medium 210. In some embodiments, thememory medium 210 may be one or more hard drives, mass storage devices,or flash drives coupled to a computer system physically by connectors orconnector cables or wirelessly by wireless communication protocols. Thekernel 208 may be stored in any computer-readable format that may beaccessed by the computer system to perform operations of methodsaccording to the present disclosure. In some implementations, thecomputer system may also be communicatively connected to the exposuretool 204 physically by connectors or connector cables or wirelessly bywireless communication protocols. The computer system may be a singlecomputer, local area networks, client-server networks, wide areanetworks, internets, a hand-held device, a wireless device, a portabledevice, a terminal, a server, a cloud computing device, a supercomputer, or a distributed system. Such a computer system may take theform of an entirely hardware embodiment, an entirely softwareembodiment, or an embodiment containing both hardware and softwareelements. By way of example, hardware generally includes at leastprocessor-capable platforms, such as client-machines (also known aspersonal computers or servers), and hand-held processing devices (suchas smart phones, personal digital assistants (PDAs), or personalcomputing devices (PCDs), for example. In addition, hardware may includeany physical device that is capable of storing machine-readableinstructions, such as a hard drive, a flash drive or other data storagedevices. Other forms of hardware include hardware sub-systems, includingtransfer devices such as modems, modem cards, ports, and port cards, forexample. In various examples, software generally includes any machinecode stored in any memory medium, such as RAM or ROM, and machine codestored on other devices (such as floppy disks, flash memory, or aCD-ROM, for example). In some embodiments, software may include sourceor object code, for example. In addition, software may encompass any setof instructions capable of being executed in a client machine or server.In embodiments of the present disclosure, after the kernel 208 is storedin the memory medium 210, the kernel 208 may be accessible, retrievable,downloadable at the computer system or other computing devicecommunicatively coupled to the computer system, operated by anauthorized user.

Referring to FIG. 1, the method 100 includes a block 114 where a firstSRAF seed map 212 is determined by convoluting the first mask function(a¹(x, y)) and the kernel 208. The first SRAF seed map 212 may beexpressed as:

Γ¹(x, y)=a ¹(x, y)⊗Ω(x, y)

As described above, the first mask function includes an X-Y component(a^(1xy)(x, y)), an X-X component (a^(1xx)(x, y))), a Y-X component(a^(1yx) (x, y)), and a Y-Y component (a^(1yy)(x, y)). The i-theigenfunction of the TCC that interacts with (a^(1xx)(x, y)) is definedas φ_(i) ^(xy)(x, y), the i-th eigenfunction of the TCC that interactswith (a^(1xy)(x, y)) is defined as φ_(i) ^(xy)(x,y), the i-theigenfunction of the TCC that interacts with (a^(1yx)(x, y)) is definedas φ_(i) ^(yx)(x, y), and the i-th eigenfunction of the TCC thatinteracts with (a^(1y)(x, y)) is defined as φ_(i) ^(yy)(x, y).Corresponding X-X kernel component (Ω^(xx)(x, y)) X-Y kernel component(Ω^(xy)(x, y)), Y-X kernel component (Ω^(yx)(x, y)), and Y-Y kernelcomponent (Ω^(yy)(x, y)) may be obtained and expressed as:

${\Omega^{xx}\left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{xx}*}\left( {0,0} \right)}{\varphi_{i}^{xx}\left( {{- x},{- y}} \right)}}} \right\rbrack}$${\Omega^{xy}\left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{xy}*}\left( {0,0} \right)}{\varphi_{i}^{xy}\left( {{- x},{- y}} \right)}}} \right\rbrack}$${\Omega^{yx}\left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{yx}*}\left( {0,0} \right)}{\varphi_{i}^{yx}\left( {{- x},{- y}} \right)}}} \right\rbrack}$${\Omega^{yy}\left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{yy}*}\left( {0,0} \right)}{\varphi_{i}^{yy}\left( {{- x},{- y}} \right)}}} \right\rbrack}$

It is noted that each of the X-X kernel component, the X-Y kernelcomponent, the Y-X kernel component, and the Y-Y kernel component isindependent from the first mask function of the first mask design. Inaddition, each of the X-X kernel component, the X-Y kernel component,the Y-X kernel component, and the Y-Y kernel component is unique to theset of exposure conditions of the exposure tool 204. It allows thekernel 208 to be stored in the memory medium 210 and reused with maskdesigns different from the first mask design 202. The X-X component, X-Ycomponent, Y-X component, and Y-Y component of the first mask functionand the X-X kernel component, X-Y kernel component, Y-X kernelcomponent, and Y-Y kernel component may be convoluted, respectively, toobtain the first SRAF seed map 212, which may also be expressed as:

Γ¹(x, y)=a ^(1xx)(x, y)⊗Ω^(xx)(x, y)+a ^(1xy)(x, y)⊗Ω^(xy)(x, y)+a^(1yx)(x, y)⊗Ω^(ys)(x, y)+a ^(1yy)(x, y)⊗Ω^(yy)(x, y)

As described above, when the first mask design 202 is or is assumed tobe an ideal mask (shown in FIG. 7, described below), the X-X, X-Y, Y-X,and YY components of the mask function are identical to one another andmay be expressed as a¹(x, y). It follows that the X-X kernel component,X-Y kernel component, Y-X kernel component, and Y-Y kernel componentthat correspond to X-X, X-Y, Y-X, and YY components of the mask functionare also identical to one another and may be expressed as Ω(x, y). As aresult, the first SRAF seed map 212 may be simplified as:

Γ¹(x, y)≈a ¹(x, y)⊗Ω(x, y)

The first SRAF seed map 212 may include an X-X map component 214 and aY-Y map component 216, as shown in FIG. 2.

Referring to FIG. 1, the method 100 includes a block 116 where furtherprocesses are performed to the first SRAF seed map 212 to obtain a firstSRAF map 218. According to the present disclosure, an SRAF seed map,such as the first SRAF seed map 212, includes information useful forSRAF placement determination. For example, an SRAF seed map includeslocal-minimum interference distribution (including intensity andcoordinates), local-maximum interference distribution (includingintensity and coordinates), and noise (including intensity andcoordinates). The local-maximum interference distribution may bereferred to as peak interference distribution and the local-minimuminterference distribution may be referred to as valley interferencedistribution. In some instances, both the peak interference distributionand the valley interference distribution include components extendingparallel to adjacent mask features and noise may include componentsextending perpendicular to adjacent mask features. Further processes aretherefore needed to eliminate or reduce a level of the noise and enhancethe signal strength of the peak interference distribution and the valleyinterference distribution. In some embodiments, high-pass filtering,bandpass filtering, or low-pass filtering may be used to remove noiseand enhance the resolution of the peak interference distribution and thevalley interference distribution. If the noise level is not reduced,SRAFs that extend perpendicular from mask features may be resulted. SuchSRAFs do not improve accuracy and may also introduce defects in theaerial image. An SRAF map, such as the first SRAF map 218, includes notonly placement positions of the SRAFs but also polygonal shapes of theSRAFs. Therefore, further processes are also needed to determine thepolygonal shapes and dimensions of the SRAFs. For example, after an SRAFseed map is filtered to remove noise and enhance resolution, polygonalshapes of suitable dimensions may be place on or around the peaks in thefiltered SRAF seed map. The process or processes to determine thepolygonal shapes and dimensions of the SRAFs may also be referred to asprocess(es) to grow SRAFs. Based on the foregoing, such furtherprocesses at block 116 include operations to reduce noise and operationsto superposition polygonal shapes based on the peak interferencedistribution the valley interference distribution.

Referring now to FIG. 1, the method 100 includes a block 118 where thefirst mask design 202 is modified according to the first SRAF map 218 toobtain a first modified mask design 220. In some embodiments, the SRAFsin the first SRAF map 218 may be placed on the first mask design 202 bysuperpositioning the first SRAF map 218 onto the first mask design 202to obtain a first modified mask design 220.

Referring now to FIGS. 1 and 2, the method 100 includes a block 120where photolithography is performed using the exposure tool 204 and afirst modified mask manufactured based on the first modified mask design220. In some embodiments, the mask house may manufacture the firstmodified mask based on the first modified mask design 220 and ship thefirst modified mask to the IC manufacturer. The IC manufacturer may thenperform photolithography using the exposure tool 204 and the firstmodified mask.

As described before, once a characteristic kernel of an exposure tool isgenerated and stored in a memory medium, and the stored kernel may beaccessed and reused with mask designs different from the first maskdesign 202 to create a different SRAF seed map. Reference is now made toFIG. 5, which illustrates a flow chart of a method 500 for semiconductorfabrication where the stored kernel is accessed and reused. Method 500will be described below in conjunction with FIG. 6, which is a schematicillustration of process steps of method 500. Additional steps can beprovided before, during, and after method 500, and some of the stepsdescribed can be moved, replaced, or eliminated for additionalembodiments of method 500.

Referring to FIGS. 5 and 6, the method 500 includes a block 502 where asecond mask design 602 that includes a second mask function (a²(x,y)) isreceived. The method 500 commences after the method 100 is concluded.Therefore, at the time the method 500 begins, the kernel 208 has alreadybeen generated and stored in the memory medium 210. The second maskdesign 602 is different from the first mask design 202 and the firstmask function (a¹(x,y)) is different from the second mask design (a²(x,y)). In some instances, the method 500 is performed by the ICmanufacturer and the second mask design 602 is received from the maskhouse. In some other instances, the second mask design 602 is providedby the mask house and the method 500 is also performed by the maskhouse.

Referring to FIGS. 5 and 6, the method 500 includes a block 504 wherethe kernel 208 stored in the memory medium 210 is retrieved. Asdescribed above, the kernel 208 is unique to the set of exposureconditions of the exposure tool 204 and after the kernel 208 isdetermined using method 100, the kernel 208 is stored in the memorymedium 210. In some embodiments represented in FIG. 6, the set ofexposure conditions of the exposure tool 204 remains the same for aphotolithography process using the second mask design 602. In theseembodiments, the kernel 208 may be retrieved at block 504 by anauthorized user and reused. As the kernel 208 is reused, there is noneed to calculate the TCC 206 and the kernel 208 again. Because thecalculation of the kernel 208 may take a substantial amount of time, thecapability to reuse the kernel may greatly reduce the turn-around-timeand conserve system resources.

Referring to FIGS. 5 and 6, the method 500 includes a block 506 where asecond SRAF seed map 612 is determined by convoluting the second maskfunction (a²(x, y)) and the stored kernel 208. Except for the use of thesecond mask function (a²(x, y)), operations at block 506 aresubstantially similar to operations at block 114. Description of thedetailed calculation is omitted for brevity.

Referring to FIGS. 5 and 6, the method 500 includes a block 508 wherefurther processes are performed to the second SRAF seed map 612 toobtain a second SRAF map 618. Except for the use of the second SRAF seedmap 612, operations at block 508 are substantially similar to operationsat block 116. Description of the detailed calculation is omitted forbrevity.

The second SRAF seed map 612 may include an X-X map component 614 and aY-Y map component 616, as shown in FIG. 6.

Referring to FIGS. 5 and 6, the method 500 includes a block 510 wherethe second mask design (a²(x,y)) is modified according to the secondSRAF map 618 to obtain a second modified mask design 620. Except for theuse of the second SRAF map 618, operations at block 510 aresubstantially similar to operations at block 118. Description of thedetailed calculation is omitted for brevity.

Referring now to FIGS. 5 and 6, the method 500 includes a block 512where photolithography is performed using the exposure tool 204 and asecond modified mask manufactured based on the second modified maskdesign 620. Except for the use of the second modified mask design 620,operations at block 510 are substantially similar to operations at block118. Description of the detailed calculation is omitted for brevity.

Non-ideal characteristic of real-world masks may also be consideredaccording to some embodiments of the present disclosure. An idealexposure configuration 700 is illustrated in FIG. 7. The ideal exposureconfiguration 700 includes an ideal mask 702, which includes aninfinitely small thickness 704 and is capable of completely blocking theincident radiation 706. Due to complete blockage of the incidentradiation 706 and the infinitely small thickness 704, a radiationamplitude 708 after the ideal mask 702 includes a step function shown inFIG. 7. Where the ideal mask 702 blocks the radiation 706, the radiationamplitude 708 drops to zero (0%). Where the ideal mask 702 allows theradiation 706 through the mask opening, the radiation amplitude 708increases to the full amplitude (100%) of the radiation 706. However, inreality, a mask has at least a finite thickness and does not havecomplete blockage of radiation. A real-world exposure configuration 800is illustrated in FIG. 8. The real-world exposure configuration 800includes a real-world mask 802, which includes a finite thickness 804and does not completely block the radiation 806. In some instances, thereal-world mask 802 may be disposed on a glass substrate 808. The finitethickness 804 and non-ideal radiation blockage capability of thereal-world mask 802 may result in a non-ideal radiation amplitude 810.These non-ideal characteristics may be summarily referred to as maskthree-dimensional (3D) effect. While only transmissive masks are shownin FIGS. 7 and 8 as example, similar ideal and non-ideal behaviors maybe observed on reflective masks as well. An ideal reflective maskincludes perfectly reflective patterns defined on a perfectly absorptivesurface. In addition, radiation is reflected only on a very top surfaceand does not penetrate into the mask. A real-world reflective maskincludes partially reflective patterns defined on a partially absorptivesurface. In terms of penetration, radiation may penetrate a depth of oneor more layers on a real-world reflective mask and may be reflected by alayer other than the topmost layer.

As radiation is an electromagnetic wave, the mask 3D effect may becalculated using the Maxwell Equations, which include the Gauss's law

$\left( {{\nabla{\cdot E}} = \frac{\rho}{ɛ_{0}}} \right),$

Gauss's law for magnetism (∇·B=0), Maxwell-Faraday equation

$\left( {{\nabla \times E} = {- \frac{\delta B}{\delta t}}} \right),$

Ampere's circuital law

$\left( {{\nabla \times B} = {{\mu_{0}\left( {J + {ɛ_{0}\frac{\delta E}{\delta t}}} \right)}.}} \right.$

In some embodiments, the mask 3D effect may be approximated using asimplified solution to the Maxwell Equations. For example, althoughSRAFs placed on a mask may contribute to the mask 3D effect too, theircontribution to the mask 3D effect is smaller than that of the main maskpattern. A simplified solution to the Maxwell Equations may drop off themask 3D effect contributed by the SRAFs. This simplification may greatlyreduce the complexity of calculation of the mask 3D effect. A solutionor a simplified solution to the Maxwell Equation may be used to modifythe mask function such that mask 3D effect is considered. The mask 3Deffect may be expressed as one or more functions M. In someimplementations, the mask 3D effect may be decomposed into an X-Xdiffraction component (M_(xx)), an X-Y diffraction component (M_(xy)), aY-X diffraction component (M_(yx)), and a Y-Y diffraction component(M_(yy)).

An example method that considers the mask 3D effect is illustrated asmethod 900. A flow chart of method 900 is illustrated in FIG. 9. Method900 will be described below in conjunction with FIG. 10, which is aschematic illustration of process steps to generate a modified maskdesign and manufacture ICs using a modified mask manufactured based onthe modified mask design. Additional steps can be provided before,during, and after method 900, and some of the steps described can bemoved, replaced, or eliminated for additional embodiments of method 900.

Referring to FIGS. 9 and 10, method 900 includes a block 902 where amask design 1002 is received. The mask design 1002 is assumed to beimplemented as a real-world mask impacted by mask 3D effect and its maskfunction may be expressed as (a^(M)(x,y)). Operations at block 902 aresubstantially similar to operations at block 102. Description of some ofthe details is omitted for brevity. The mask function (a^(M)(x,y))includes an X-X component (a^(Mxx)(x, y)), an X-Y component (a^(Mxy)(x,y)), a Y-X component (a^(Myx) (x, y)), and a Y-Y component (a^(Myy)(x,y)). The X-X component (a^(Mxx)(x, y)), the X-Y component (a^(Mxy)(x,y)), the Y-X component (a^(Myx) (x, y)), and the Y-Y component(a^(Myy)(x, y)) may be obtained by taking Fourier Transform of thecorresponding diffraction components:

a ^(Mxx)(x, y)=FT(M _(xx)),

a ^(Mxy)(x, y)=FT(M _(xy)),

a ^(Myx)(x, y)=FT(M _(yx)),

a ^(Myy)(x, y)=FT(M _(yy)).

Unlike their counterparts for an assumed ideal mask, the X-X component,X-Y component, Y-X component, and Y-Y component of the mask functiona^(M)(x, y) are not identical to one another and are to be separatedconsidered. Because the X-X component, X-Y component, Y-X component, andY-Y component of the mask function a^(M)(x, y) are not identical to oneanother, the corresponding kernel components Ω^(Mxx)(x, y), Ω^(Mxy)(x,y), Ω^(Myx)(x, y), and Ω^(Myy)(x, y) are not identical to one anotherand should be considered separately.

Referring now to FIGS. 9 and 10, method 900 includes a block 904 wherean exposure tool 1004 is provided. Operations at block 904 aresubstantially similar to operations at block 104. Description of thedetails is omitted for brevity.

Referring now to FIGS. 9 and 10, method 900 includes a block 906 where atransmission cross coefficient (TCC) 1006 of the exposure tool 1004 isdetermined based on the set of the exposure conditions. Operations atblock 904 are substantially similar to operations at block 104.Description of the details is omitted for brevity.

Referring now to FIGS. 9 and 10, method 900 includes a block 908 where atransmission cross coefficient (TCC) 1006 is decomposed into a pluralityorders of eigenvalues and a plurality orders of eigenfunctions.

Referring now to FIGS. 9 and 10, method 900 includes a block 910 where akernel 1008 is calculated based on the plurality orders of eigenvalues(λ_(i)) and the plurality orders of eigenfunctions (φ_(i)), and the mask3D diffraction components 1005. Except for the integration of thediffraction components, operations at block 910 are substantiallysimilar to those at block 110. Descriptions of some of the details isomitted for brevity. As described above, the kernel 1008 includes X-Xkernel component (Ω^(Mxx)(x,y)) X-Y kernel component (Ω^(Mxy)(x, y)),Y-X kernel component Ω^(Myx)(x, y), and Y-Y kernel component Ω^(Myy) (x,y) that may be separately considered and calculated.

Referring now to FIGS. 9 and 10, method 900 includes a block 912 wherethe kernel 1008 is stored in a memory medium 1010. Operations at block912 are substantially similar to operations at block 112. Description ofthe details is omitted for brevity. In embodiments of the presentdisclosure, after the kernel 1008 is stored in the memory medium 1010,the kernel 1008 may be accessible, retrievable, downloadable at thecomputer system or other computing device communicatively coupled to thecomputer system, operated by an authorized user.

Referring now to FIGS. 9 and 10, method 900 includes a block 914 wherean SRAF seed map 1012 is determined by convoluting the mask function(a^(M)(x, y)) and the kernel 1008. Except for the integration of themask 3D diffraction components, operations at block 914 aresubstantially similar to those at block 114. Descriptions of some of thedetails is omitted for brevity. The SRAF seed map 1012 may be expressedas:

Γ^(M)(x, y)=a ^(Mxx)(x, y)⊗Ω^(Mxx)(x, y)+a ^(Mxy)(x, y)⊗Ω^(Mxy)(x, y)+a^(Myx)(x, y)⊗ΩM ^(yx)(x, y)+a ^(Myy)(x, y)⊗Ω^(Myy)(x, y)

It has been observed that the X-X kernel component and the Y-Y kernelcomponent may be the predominant kernel components. In some embodiments,the X-Y kernel component and the Y-X kernel component may be omitted tofurther improve the turn-around-time and the SRAF seed map 1012 may beapproximated as:

Γ^(M)(x, y)=ba ^(Mxx)(x, y)⊗Ω^(Mxx)(x, y)+a ^(Myy)(x, y)⊗Ω^(Myy)(x, y)

The SRAF seed map 1012 may include an X-X map component 1014 and a Y-Ymap component 1016, as shown in FIG. 10.

Referring now to FIGS. 9 and 10, method 900 includes a block 916 wherefurther processes are performed to the SRAF seed map 1012 to obtain anSRAF map 1018. Operations at block 916 are substantially similar tothose at block 116. Descriptions of some of the details is omitted forbrevity.

Referring now to FIGS. 9 and 10, method 900 includes a block 918 wherethe mask design 1002 is modified according to the SRAF map 1018 toobtain a modified mask design 1020 (A′(x, y)). Operations at block 918are substantially similar to those at block 118. Descriptions of some ofthe details is omitted for brevity.

Referring now to FIGS. 9 and 10, the method 900 includes a block 920where photolithography is performed using the exposure tool 1004 and amodified mask manufactured based on the modified mask design 1020.Operations at block 920 are substantially similar to those at block 120.Descriptions of some of the details is omitted for brevity.

While not separately illustrated in a separate flow chart, the kernel1008 may be stored in the memory medium 1010 and reused in conjunctionwith a subsequent mask design different from the mask design (a^(M)(x,y)), provided that the subsequent mask design includes a similar maskcomposition and a similar mask thickness. The reuse of the kernel 1008may greatly reduce the turn-around-time without compromising accuracy ofthe SRAF placement.

The embodiments of the present disclosure offer advantages over existingart, though it is understood that other embodiments may offer differentadvantages, not all advantages are necessarily discussed herein, andthat no particular advantage is required for all embodiments. Byincluding all order of the eigenvalues and eigenfunctions of TCC in thecalculation of a kernel, methods according to embodiments of the presentdisclosure consider exposure conditions of the exposure tool, includingillumination intensity of the exposure tool, a numerical aperture of theexposure tool, a depth of focus (DOF), a thickness of a resist stack tobe patterned, or a range of an aberration. In addition, methods of thepresent disclosure may include diffraction components to addresspolarization due to mask three-dimensional (3D) effect. Because of theconsideration of the exposure conditions and the mask 3D effect, methodsof the present disclosure have better accuracy than conventionalrule-based SRAF placement techniques and IML techniques. Moreover, whenmethods of the present disclosure are used, a kernel generated based ona set of exposure conditions of an exposure tool may be stored andreused whenever a new mask design is introduced. The reuse of the kernelmay greatly reduce the turn-around-time to a level similar to that ofthe rule-based techniques, which is a fraction of the turn-around-timeof ILT.

Thus, one of the embodiments of the present disclosure described amethod for fabricating a semiconductor device. The method includesreceiving a first mask design including a first mask function,determining a transmission cross coefficient (TCC) of an exposure tool,decomposing the TCC into a plurality orders of eigenvalues and aplurality orders of eigenfunctions, calculating a kernel based on theplurality orders of eigenvalues and the plurality orders ofeigenfunctions, and determining a first sub-resolution assist feature(SRAF) seed map by convoluting the first mask function and the kernel.

In some embodiments, the first SRAF seed map includes coordinates ofpeak positions for placement of a plurality of SRAFs. In someembodiments, the method further includes processing the first SRAF seedmap to obtain a first SRAF map, modifying the first mask designaccording to the first SRAF map to obtain a first modified mask design,and performing photolithography using the exposure tool and the firstmodified mask design. In some implementations, the first SRAF mapincludes a plurality of SRAFs, and a polygonal shape of each of theplurality of SRAFs. In some instances, the method further includesstoring the kernel in a memory medium, receiving a second mask designincluding a second mask function different from the first mask function,retrieving the kernel stored in the memory medium, determining a secondSRAF seed map by convoluting the second mask function and the storedkernel, processing the second SRAF seed map to obtain a second SRAF map,modifying the second mask design according to the second SRAF map toobtain a second modified mask design, and performing photolithographyusing the exposure tool and the second modified mask design. In someimplementations, the exposure tool includes an extreme ultraviolet (EUV)exposure tool or a deep ultraviolet (DUV) exposure tool. In someinstances, the TCC includes information about an illumination intensityof the exposure tool, a numerical aperture of the exposure tool, athickness of a resist stack to be patterned, or a range of anaberration.

In another of the embodiments, a method of semiconductor devicefabrication is provided. The method includes receiving a first maskdesign including a first mask function (a (x,y)), providing an exposuretool that includes a set of exposure conditions, determining atransmission cross coefficient (TCC) of the exposure tool based on theset of exposure conditions, decomposing the TCC into a plurality ordersof eigenvalues (λ_(i)) and a plurality orders of eigenfunctions(φ_(i)(x,y)), calculating a kernel (Ω(x.y)) based on the followingmathematical formulae:

${\Omega \left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{*}\left( {0,0} \right)}{\varphi_{i}\left( {{- x},{- y}} \right)}}} \right\rbrack}$

,and determining a first sub-resolution assist feature (SRAF) seed map(Γ(x,y)) by convoluting the kernel and the first mask function using thefollowing mathematical formula:

Γ(x, y)=a( x, y)⊗Ω(x, y).

In some embodiments, the method further includes determining a firstsub-resolution assist feature (SRAF) seed map (Γ(x,y)) by convolutingthe kernel and the first mask function using the following mathematicalformula:

Γ(x,y)=a(x, y)⊗Ω(x, y).

The first mask design is assumed to be implemented as an ideal mask. Insome embodiments, the first mask function includes an X-X component(a^(xx)(x,y)), an X-Y component (a^(xy)(x,y)), a Y-X component(a^(yx)(x,y)), and a Y-Y component (a^(yy)(x,y)). The plurality ordersof eigenvalues include a first plurality orders of X-X interactioneigenfunctions (φ_(i) ^(xx)(x,y)), a second plurality orders of X-Yinteraction eigenfunctions (φ_(i) ^(xy) (x,y)), a third plurality ordersof Y-X interaction eigenfunctions (φ_(i) ^(yx)(x,y)), and a fourthplurality orders of Y-Y interaction eigenfunctions (φ_(i) ^(yy) (x,y)).The kernel includes an X-Y kernel component (Ω^(xy) (x.y)), a Y-X kernelcomponent (φ_(i) ^(yy)(x.y)), and a Y-Y kernel component (Ω^(yy) (x.y))respectively expressed by the following mathematical formulae:

${{\Omega^{XX}\left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{XX}*}\left( {0,0} \right)}{\varphi_{i}^{XX}\left( {{- x},{- y}} \right)}}} \right\rbrack}},{{\Omega^{XY}\left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{XY}*}\left( {0,0} \right)}{\varphi_{i}^{XY}\left( {{- x},{- y}} \right)}}} \right\rbrack}},{{\Omega^{YX}\left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{YX}*}\left( {0,0} \right)}{\varphi_{i}^{YX}\left( {{- x},{- y}} \right)}}} \right\rbrack}},{{{and}{\Omega^{YY}\left( {x,y} \right)}} = {{{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{YY}*}\left( {0,0} \right)}{\varphi_{i}^{YY}\left( {{- x},{- y}} \right)}}} \right\rbrack}.}}$

In some embodiments, the method further includes determining a firstsub-resolution assist feature (SRAF) seed map (F(x,y)) by convolutingthe kernel and the first mask function using the following mathematicalformula:

Γ(x,y)=a ^(xx)(x,y)⊗Ω^(xx)(x, y)+a ^(xy) (x, y)⊗Ω^(xy) (x, y)+a ^(yx)(x,y)⊗Ω^(yx)(x, y)+a ^(yy)(x, y) ⊗Ω^(yy)(x, y),

wherein the first mask design is assumed to be implemented as areal-world mask. In some embodiments, the set of exposure conditionsincludes an illumination intensity of the exposure tool, a numericalaperture of the exposure tool, a thickness of a resist stack to bepatterned, or a range of an aberration. In some implementations, themethod further includes storing the kernel in a memory medium. In someembodiments, the method further includes processing the first SRAF seedmap to obtain a first SRAF map, modifying the first mask designaccording to the first SRAF map to obtain a first modified mask design,and performing photolithography using the exposure tool and the firstmodified mask design. In some instances, the processing of the firstSRAF seed map includes filtering the first SRAF seed map to removenoise, resulting in a filtered first SRAF seed map and fitting polygonalshapes onto the filtered first SRAF seed map.

In yet other embodiments, a method of semiconductor device fabricationis provided. The method includes receiving a first mask design includinga first mask function (a(x, y)), the first mask function (a(x,y))including an X-X component (a^(xx)(x,y)) and a Y-Y component(a^(yy)(x,y)), providing an exposure tool that includes a set ofexposure conditions, determining a transmission cross coefficient (TCC)of the exposure tool based on the set of exposure conditions,decomposing the TCC into a plurality orders of eigenvalues (λ_(i)), afirst plurality orders of X-X interaction eigenfunctions (φ_(i)^(xx)(x,y)), a second plurality orders of Y-Y interaction eigenfunctions(φ_(i) ^(yy) (x,y)), calculating an X-X kernel component (Ω^(xx) (x.y))and a Y-Y kernel component (Ω^(yy) (x.y)) based on the followingmathematical formulae:

${{\Omega^{XX}\left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{XX}*}\left( {0,0} \right)}{\varphi_{i}^{XX}\left( {{- x},{- y}} \right)}}} \right\rbrack}},{and}$${{\Omega^{YY}\left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{YY}*}\left( {0,0} \right)}{\varphi_{i}^{YY}\left( {{- x},{- y}} \right)}}} \right\rbrack}},$

anddetermining a first sub-resolution assist feature (SRAF) seed map(Γ(x,y)) by convoluting the kernel and the first mask function using thefollowing mathematical formula:

Γ(x,y)=a ^(xx)(x, y)⊗Ω^(xx)(x, y)+a ^(yy)(x, y)⊗Ω^(yy)(x, y).

In some embodiments, the exposure tool includes a pupil. The X-X kernelcomponent includes a simulated interaction between an X-polarizedradiation on the first mask design and an X-polarized radiation on thepupil and the Y-Y kernel component includes a simulated interactionbetween a Y-polarized radiation on the first mask design and aY-polarized radiation on the pupil. In some implementations, the methodfurther includes processing the first SRAF seed map to obtain a firstSRAF map, modifying the first mask design according to the first SRAFmap to obtain a first modified mask design, and performingphotolithography using the exposure tool and the first modified maskdesign. In some instances, the processing of the first SRAF seed mapincludes filtering the first SRAF seed map to remove noise, resulting ina filtered first SRAF seed map, and fitting polygonal shapes onto thefiltered first SRAF seed map. In some embodiments, the exposure toolincludes an extreme ultraviolet (EUV) exposure tool or a deepultraviolet (DUV) exposure tool.

The foregoing outlines features of several embodiments so that thoseskilled in the art may better understand the aspects of the presentdisclosure. Those skilled in the art should appreciate that they mayreadily use the present disclosure as a basis for designing or modifyingother processes and structures for carrying out the same purposes and/orachieving the same advantages of the embodiments introduced herein.Those skilled in the art should also realize that such equivalentconstructions do not depart from the spirit and scope of the presentdisclosure, and that they may make various changes, substitutions, andalterations herein without departing from the spirit and scope of thepresent disclosure.

1. A method of semiconductor device fabrication, comprising: receiving afirst mask design comprising a first mask function; determining atransmission cross coefficient (TCC) of an exposure tool; decomposingthe TCC into a plurality orders of eigenvalues and a plurality orders ofeigenfunctions; calculating a kernel based on the plurality orders ofeigenvalues and the plurality orders of eigenfunctions; determining afirst sub-resolution assist feature (SRAF) seed map by convoluting thefirst mask function and the kernel; storing the kernel in a memorymedium; receiving a second mask design comprising a second mask functiondifferent from the first mask function; retrieving the kernel stored inthe memory medium; determining a second SRAF seed map by convoluting thesecond mask function and the stored kernel; processing the second SRAFseed map to obtain a second SRAF map; modifying the second mask designaccording to the second SRAF map to obtain a second modified maskdesign; and performing photolithography using the exposure tool and thesecond modified mask design.
 2. The method of claim 1, wherein the firstSRAF seed map comprises coordinates of peak positions for placement of aplurality of SRAFs.
 3. The method of claim 1, further comprising:processing the first SRAF seed map to obtain a first SRAF map; modifyingthe first mask design according to the first SRAF map to obtain a firstmodified mask design; and performing photolithography using the exposuretool and the first modified mask design.
 4. The method of claim 3,wherein the first SRAF map comprises: a plurality of SRAFs; and apolygonal shape of each of the plurality of SRAFs.
 5. (canceled)
 6. Themethod of claim 1, wherein the exposure tool comprises an extremeultraviolet (EUV) exposure tool or a deep ultraviolet (DUV) exposuretool.
 7. The method of claim 1, wherein the TCC comprises informationabout an illumination intensity of the exposure tool, a numericalaperture of the exposure tool, a thickness of a resist stack to bepatterned, or a range of an aberration.
 8. A method of semiconductordevice fabrication, comprising: receiving a first mask design comprisinga first mask function (a (x,y)); providing an exposure tool thatincludes a set of exposure conditions; determining a transmission crosscoefficient (TCC) of the exposure tool based on the set of exposureconditions; decomposing the TCC into a plurality orders of eigenvalues(λ_(i)) and a plurality orders of eigenfunctions (φ_(i)(x,y));calculating a kernel (Ω(x,y)) based on the following mathematicalformulae:${\Omega \left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{*}\left( {0,0} \right)}{\varphi_{i}\left( {{- x},{- y}} \right)}}} \right\rbrack}$; and determining a first sub-resolution assist feature (SRAF) seed map(Γ(x,y)) by convoluting the kernel and the first mask function using thefollowing mathematical formula:Γ(x,y)=a(x,y)⊗Ω(x,y).
 9. The method of claim 8, further comprising:determining a first sub-resolution assist feature (SRAF) seed map(Γ(x,y)) by convoluting the kernel and the first mask function using thefollowing mathematical formula:Γ(x,y)=a(x,y)⊗Ω(x,y), wherein the first mask design is assumed to beimplemented as an ideal mask.
 10. The method of claim 8, wherein thefirst mask function includes an X-X component (a^(xx)(x,y)), an X-Ycomponent (a^(xy)(x,y)), a Y-X component (a^(yx)(x,y)), and a Y-Ycomponent (a^(yy)(x,y)), wherein the plurality orders of eigenvaluesinclude a first plurality orders of X-X interaction eigenfunctions(φ_(i) ^(xx) (x,y)), a second plurality orders of X-Y interactioneigenfunctions (φ_(i) ^(xy) (x,y)), a third plurality orders of Y-Xinteraction eigenfunctions (φ_(i) ^(yx) (x,y)), and a fourth pluralityorders of Y-Y interaction eigenfunctions (®_(i) ^(yy) (x,y)), whereinthe kernel includes an X-Y kernel component (Ω^(xy) (x.y)), a Y-X kernelcomponent (Ω^(yx) (x.y)), and a Y-Y kernel component (Ω^(yy) (x.y))respectively expressed by the following mathematical formulae:${{\Omega^{XX}\left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{XX}*}\left( {0,0} \right)}{\varphi_{i}^{XX}\left( {{- x},{- y}} \right)}}} \right\rbrack}},{{\Omega^{XY}\left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{XY}*}\left( {0,0} \right)}{\varphi_{i}^{XY}\left( {{- x},{- y}} \right)}}} \right\rbrack}},{{\Omega^{YX}\left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{YX}*}\left( {0,0} \right)}{\varphi_{i}^{YX}\left( {{- x},{- y}} \right)}}} \right\rbrack}},{{{and}{\Omega^{YY}\left( {x,y} \right)}} = {{{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{YY}*}\left( {0,0} \right)}{\varphi_{i}^{YY}\left( {{- x},{- y}} \right)}}} \right\rbrack}.}}$11. The method of claim 10, further comprising: determining a firstsub-resolution assist feature (SRAF) seed map (Γ(x,y)) by convolutingthe kernel and the first mask function using the following mathematicalformula:Γ(x,y)=a ^(xx)(x,y)⊗Ω^(xx) (x,y)+a ^(xy)(x,y)⊗Ω^(xy)(x,y)+a^(yx)(x,y)⊗Ω^(yx)(x,y)+a ^(yy)(x,y)⊗Ω^(yy)(x,y), wherein the first maskdesign is assumed to be implemented as a real-world mask.
 12. The methodof claim 8, wherein the set of exposure conditions comprises anillumination intensity of the exposure tool, a numerical aperture of theexposure tool, a thickness of a resist stack to be patterned, or a rangeof an aberration.
 13. The method of claim 8, further comprising storingthe kernel in a memory medium.
 14. The method of claim 8, furthercomprising: processing the first SRAF seed map to obtain a first SRAFmap; modifying the first mask design according to the first SRAF map toobtain a first modified mask design; and performing photolithographyusing the exposure tool and the first modified mask design.
 15. Themethod of claim 14, wherein the processing of the first SRAF seed mapcomprises: filtering the first SRAF seed map to remove noise, resultingin a filtered first SRAF seed map; and fitting polygonal shapes onto thefiltered first SRAF seed map.
 16. A method of semiconductor devicefabrication, comprising: receiving a first mask design comprising afirst mask function (a(x,y)), the first mask function (a(x,y)) includingan X-X component (a^(xx)(x,y)) and a Y-Y component (a^(yy)(x.y));providing an exposure tool that includes a set of exposure conditions;determining a transmission cross coefficient (TCC) of the exposure toolbased on the set of exposure conditions; decomposing the TCC into aplurality orders of eigenvalues (λ_(i)), a first plurality orders of X-Xinteraction eigenfunctions (φ_(i) ^(xx) (x,y)), a second pluralityorders of Y-Y interaction eigenfunctions (φ_(i) ^(yy) (x,y));calculating an X-X kernel component (Ω^(xx) (x.y)) and a Y-Y kernelcomponent (Ω^(yy) (x.y)) based on the following mathematical formulae:${{\Omega^{XX}\left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{XX}*}\left( {0,0} \right)}{\varphi_{i}^{XX}\left( {{- x},{- y}} \right)}}} \right\rbrack}},{and}$${{\Omega^{YY}\left( {x,y} \right)} = {{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{{YY}*}\left( {0,0} \right)}{\varphi_{i}^{YY}\left( {{- x},{- y}} \right)}}} \right\rbrack}};$and determining a first sub-resolution assist feature (SRAF) seed map(Γ(x,y)) by convoluting the kernel and the first mask function using thefollowing mathematical formula:Γ(x,y)=a ^(xx)(x,y)⊗Ω^(xx)(x,y)+a ^(yy)(x,y)⊗Ω^(yy)(x,y).
 17. The methodof claim 16, wherein the exposure tool comprises a pupil, wherein theX-X kernel component comprises a simulated interaction between anX-polarized radiation on the first mask design and an X-polarizedradiation on the pupil, wherein the Y-Y kernel component comprises asimulated interaction between a Y-polarized radiation on the first maskdesign and a Y-polarized radiation on the pupil.
 18. The method of claim16, further comprising: processing the first SRAF seed map to obtain afirst SRAF map; modifying the first mask design according to the firstSRAF map to obtain a first modified mask design; and performingphotolithography using the exposure tool and the first modified maskdesign.
 19. The method of claim 18, wherein the processing of the firstSRAF seed map comprises: filtering the first SRAF seed map to removenoise, resulting in a filtered first SRAF seed map; and fittingpolygonal shapes onto the filtered first SRAF seed map.
 20. The methodof claim 16, wherein the exposure tool comprises an extreme ultraviolet(EUV) exposure tool or a deep ultraviolet (DUV) exposure tool.
 21. Themethod of claim 1, wherein the plurality orders of eigenvalues aredenoted as λ_(i), wherein the plurality orders of eigenfunctions aredenoted as φ_(i), wherein the kernel is denoted as Ω(x,y), wherein thecalculating of the kernel comprises use of the following mathematicalformula:${\Omega \left( {x,y} \right)} = {{{Re}\left\lbrack {\sum\limits_{i = 1}^{N}{\lambda_{i}{\varphi_{i}^{*}\left( {0,0} \right)}{\varphi_{i}\left( {{- x},{- y}} \right)}}} \right\rbrack}.}$